Cat 6 cable is Ethernet cabling, not FireWire cabling, so rebooting into FireWire mode is not going to work. Just plug each end of the Ethernet cable into the Ethernet port of each Mac (port with 2 arrows separated by dotted line).
On the G5 iMac, go to the Apple Menu > System Preferences > Network. Click on Show: and select Built-in Ethernet. Select the AppleTalk tab and check Make AppleTalk Active.
On the Blueberry G3, go to the Apple Menu > Control Panels > File Sharing. Click on Start under File Sharing and then Check the box Enable File Sharing clients to connect over TCP/IP. Close the control panel. Now go to the Apple Menu > Control Panels > AppleTalk. Click on Connect Via: and select Ethernet. Close the control panel. Restart your old Mac to make sure the settings changed take effect.
Now go back to your G5 iMac and while the Finder is active and with a Finder window open, click on Network in the Finder Window Sidebar. Your old mac should be visible. You can now copy your old System, Apps and Files over.
OS 9 IS history for most Mac users. Once you get up and running on X you will understand why. You can run the majority of 9 apps in Classic mode (9 booted inside X) without a hitch, but your mileage may vary. Aside from having to run QuarkXPress 5 (not X native and I refuse to give them anymore money), Illustrator 9 and Photoshop 7 (for plug-ins not updated yet or discontinued altogether), and the occasional old fave game, 99% of my time is spent in X and I am happier for it. You don't have to upgrade everything at once, or at all if you choose, but things like Photoshop CS (version 8 by the way, compared to your version 4) and Illustrator CS (version 11, compared to your 8 which was one of the worst versions of Illustrator ever) will be a welcome change...
Good luck. And welcome to the future...